Airy phase functions

It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in the commonly-occurring case in which the equation has turning points. We resolve this difficulty by introducing a generalized phase function method for second order linear ordinary differential equations with turning points. More explicitly, we give an efficient numerical algorithm for computing an Airy phase function which efficiently represents the solutions of such an equation. The running time of our algorithm is independent of the magnitude of the logarithmic derivatives of the equation’s solutions, which is a measure of their rate of variation that generalizes the notion of frequency to functions which are rapidly varying but not necessarily oscillatory. Once the Airy phase function has been constructed, any reasonable initial or boundary value problem can be readily solved and, unlike step methods that only generate the values of a rapidly-varying solution at the nodes of a sparse discretization grid that is insufficient for interpolation, the output of our scheme allows for the evaluation of the solution at any point in the equation’s domain. We rigorously justify our approach by proving not only the existence of slowly-varying Airy phase functions, but also the convergence of our numerical method. Moreover, we present the results of extensive numerical experiments demonstrating the efficacy of our algorithm.